The main idea is that under the hypotheses \((H_{1})\) -\((H_{4})\)  we transform Sobolev type fractional multi-term evolution equation with linear operators (REF ) to fractional-order evolution equation with multi-orders and linear bounded operators (REF ). Secondly, we solve fractional evolution equation with nonpermutable linear bounded operators by using Laplace transform technique which is used as a necessary tool for solving and analyzing fractional-order differential equations and systems in [1]},[2]},[3]},[4]}.
Then we propose exact analytical representation of a mild solution of (REF ) and (REF ), respectively with the help of new defined Mittag-Leffler function which is expressed via linear bounded operators by removing the exponential boundedness of a forced term \(g(\cdot )\)  and \(\left( {C}{}{D}_{0+}^{\beta }x\right)(\cdot )\)  for \(\beta \in (0,1]\)  (or \(\left( {C}{}{D}_{0+}^{\alpha }x\right)(\cdot )\)  for \(\alpha \in (1,2])\)  in both cases: with nonpermutable and permutable linear operators \(A,B\in {B}(Y)\) .
